Critical behavior of the planar magnet model in three dimensions

a r X i v :c o n d -m a t /9811276v 1 [c o n d -m a t .s t a t -m e c h ] 18 N o v 1998Critical behavior of the planar magnet model in three dimensions Kwangsik Nho and Efstratios Manousakis Department of Physics and Center for Research and Technology,Florida State University,Tallahassee,Florida 32306-4350(February 1,2008)We study the critical behavior of the three-dimensional planar magnet model in which each spin is considered to have three components of which only the x and y components are coupled.We use a hybrid Monte Carlo al-gorithm in which a single-cluster update is combined with the over-relaxation and Metropolis spin re-orientation algorithm.Periodic boundary conditions were applied in all directions.We have calculated the fourth-order cumulant in finite size lattices using the single-histogram re-weighting ing finite-size scaling theory,we obtained the critical temperature which is very different from that of the usual XY model.At the critical temperature,we cal-culated the susceptibility and the magnetization on lattices of size up to ing finite-size scaling theory we accurately determine the critical exponents of the model and find that ν=0.670(7),γ/ν=1.9696(37),and β/ν=0.515(2).Thus,we conclude that the model belongs to the same universality class with the XY model,as expected.64.60.Fr,67.40.-w,67.40.Kh I.INTRODUCTION Our understanding of critical phenomena has been significantly advanced with the devel-opment of the renormalization-group (RG)theory [1].The RG theory predicts relationships between groups of exponents and that there is a universal behavior.In a second order phase transition,the correlation length ξdiverges as the critical point is approached,and so the details of the microscopic Hamiltonian are unimportant for the critical behavior.Allmembers of a given universality class have identical critical behavior and critical exponents.The three-dimensional classical XY model is relevant to the critical behavior of many physical systems,such as superfluid4He,magnetic materials and the high-Tc superconduc-tors.In the pseudospin notation,this model is defined by the Hamiltonian(S x i S x j+S y i S y j),(1)H=−J<ij>where the summation is over all nearest neighbor pairs of sites i and j on a simple cubic lattice.In this model one considers that the spin has two components, S i=(S x i,S y i)and S x2i+S y2i=1.In this paper we wish to consider a three component local spin S i=(S x i,S y i,S z i)and the same Hamiltonian as given by Eq.(1)(namely,with no coupling between the z-components of the spins)in three dimensions.Even though the Hamiltonian is the same,namely,there is no coupling between the z-component of the spins,the constrain for each spin is(S x i)2+ (S y i)2+(S z i)2=1,which implies that the quantity(S x i)2+(S y i)2isfluctuating.In order to be distinguished from the usual XY model,the name planar magnet model will be adopted for this model.The reason for our desire to study this model is that it is related directly to the so-called model-F[2]used to study non-equilibrium phenomena in systems,such as superfluids,with a two-component order parameter and a conserved current.In the planar magnet model, the order parameter is not a constant of the motion.A constant of the motion is the z component of the magnetization.Thus,there is an important relationship between the order parameter and the z component of magnetization,which is expressed by a Poisson-bracket relation[2].This equation is crucial for the hydrodynamics and the critical dynamics of the system.One therefore needs tofind out the critical properties of this model in order to study non-equilibrium properties of superfluids or other systems described by the model F.In future work,we shall use model-F to describe the dynamical critical phenomena of superfluid helium.Before such a project is undertaken,the static critical properties of the planar magnet model should be investigated accurately.Although the static properties of the XY model with S i=(S x i,S y i)have been investi-gated by a variety of statistical-mechanical methods[3–11],the system with S i=(S x i,S y i,S z i) has been given much less attention.So far the critical behavior of this model has been stud-ied by high temperature expansion[12]and Monte Carlo(MC)simulation methods[13,14]. High temperature expansion provides the value for the critical temperature and the critical exponents.In these recent MC calculations[13,14],only the critical temperature is deter-mined.These MC calculations were carried out on small size systems and thus only rough estimates are available.In this paper we study the three-dimensional planar magnet model using a hybrid Monte Carlo method(a combination of the cluster algorithm with over-relaxation and Metropolis spin re-orientation algorithm)in conjuction with single-histogram re-weighting technique andfinite-size scaling.We calculate the fourth order cumulant,the magnetization,and the susceptibility(on cubic lattices L×L×L with L up to42)and from theirfinite-size scaling behavior we determine the critical properties of the planar magnet model accurately.II.PHYSICAL QUANTITIES AND MONTE CARLO METHODLet usfirst summarize the definitions of the observables that are calculated in our sim-ulation.The energy density of our model is given by1<e>=E/V=,(3)3<m2>2where m=1K c.In the scaling region close to the critical coupling,the fourth-order cumulant U L(K)as function of K for different values of L are lines which go through the same point.The magnetic susceptibility per spinχis given byχ=V K(<m2>−< m>2),(4)where m is the magnetization vector per spin.The three-dimensional planar magnet model with ferromagnetic interactions J>0has a second-order phase transition.In simulations of systems near a second-order phase transi-tion,a major difficulty arises which is known as critical slowing down.The critical slowing down can be reduced by using several techniques and what we found as optimal for our case was to use the hybrid Monte Carlo algorithm as described in Ref.[16].Equilibrium configurations were created using a hybrid Monte Carlo algorithm which combines cluster updates of in-plane spin components[17]with Metropolis and over-relaxation[18]of spin re-orientations.After each single-cluster update,two Metropolis and eight over-relaxation sweeps were performed[16].The K dependence of the fourth-order cumulant U L(K)was determined using the single-histogram re-weighting method[19].This method enables us to obtain accurate thermodynamic information over the entire scaling region using Monte Carlo simulations performed at only a few different values of K.We have performed Monte Carlo simulation on simple cubic lattices of size L×L×L with6≤L≤42using periodic boundary conditions applied in all directions and106MC steps.We carried out of the order of10000thermalization steps and of the order of20000measurements.After we estimated the critical coupling K c,we computed the magnetization and the magnetic susceptibility at the critical coupling K c.III.RESULTS AND DISCUSSIONIn this section,wefirst have to determine the critical coupling K c,and then to examine the static behavior around K c.Binder’s fourth-order cumulant[15]U L(K)is a convenientquantity that we use in order to estimate the critical coupling K c and the correlation length exponentν.Near the critical coupling K c,the cumulant is expanded asTU L=U∗+U1L1/ν(1−L and L ′=bL and then find the location of the intersection of two different curves U L and U L ′.As a result of the residual corrections to the finite size scaling [15],the locations depend on the scale factor b =L ′/L .We used the crossing points of the L =12,14,and,16curves with all the other ones with higher L ′value respectively.Hence we need to extrapolate the results of this method for (ln b )−1−→0using (U bL /U L )T =T c =1.In Fig.2we show theestimate for the critical temperature T c .Our final estimate for T c isT c =1.5518(2),K c =0.6444(1).(6)For comparison,the previous estimates are T c =1.54(1)[13,14]obtained using Monte Carlo simulation and T c =1.552(3)[12]obtained using high-temperature series.The latter result obtained with an expansion is surprisingly close to ours.0.0 2.0 4.06.08.010.01/log(b)1.54901.55001.55101.55201.55301.5540L=12L=14L=16FIG.2.Estimates for T c plotted versus inverse logarithm of the scale factor b=L ′/L .The extrapolation leads to an estimate of T c =1.5518(2).In order to extract the critical exponent ν,we performed finite-size scaling analysis of the slopes of U L versus L near our estimated critical point K c .In the finite-size scaling region,the slope of the cumulant at K c varies with system size like L 1/ν,dU LIn Fig.3we show results of a finite-size scaling analysis for the slope of the cumulant.We obtained the value of the static exponent ν,ν=0.670(7).(8)For comparison,the field theoretical estimate [3]is ν=0.669(2)and a recent experimental measurement gives ν=0.6705(6)[20].log(L)1.02.03.04.0l o g (d U /d K )FIG.3.Log-log plot of the slopes of U near the crossing point versus L .The slope gives an estimate for the critical exponent ν=0.670(7).In order to obtain the value of the exponent ratio γ/ν,we calculated the magnetic susceptibility per spin χat the critical coupling K c .The finite-size behavior for χat the critical point isχ∼L γ/ν.(9)Fig.4displays the finite-size scaling of the susceptibility χcalculated at K c =0.6444.¿From the log-log plot we obtained the value of the exponent ratio γ/ν,γ/ν=1.9696(37).(10)¿From the hyperscaling relation,dν=γ+2β,we get the exponent ratio β/ν,β/ν=0.515(2).(11)2.0 2.53.03.54.0log(L)4.05.06.07.0l o g (χ)FIG.4.Log-log plot of the susceptibility versus the lattice size L at the critical coupling K c =0.6444.The slope gives an estimate for the critical exponent γ/ν=1.9696(37).The equilibrium magnetization m at K c should obey the relationm ∼L −β/ν(12)for sufficiently larger L .In Fig.5we show the results of a finite-size scaling analysis for the magnetization m .We obtain the value of the exponent ratio β/ν,β/ν=0.515(2).(13)This result agrees very closely to that of Eq.(11)obtained from the susceptibility and the fourth-order cumulant.In conclusion,we determined the critical temperature and the exponents of the planar magnet model with three-component spins using a high-precision MC method,the single-histogram method,and the finite-size scaling theory.Our simulation results for the critical coupling and for the critical exponents are K c =0.6444(1),ν=0.670(7),γ/ν=0.9696(37),and β/ν=0.515(2).Our calculated values for the critical temperature and critical exponents are significantly more accurate that those previously parison of our results with results of MC studies of the 3D XY model with two-component spins [7,9–11]shows thatboth the system with S i =(S x i ,S y i )and the planar magnet system with S i =(S x i ,S y i ,S z i)belong to the same universality class.log(L)-2.0-1.8-1.6-1.4-1.2l o g (m )FIG.5.Log-log plot of the magnetization versus the lattice size L at the critical coupling K c =0.6444.The slope gives an estimate for the critical exponent β/ν=0.515(3).Lm 120.26195(55)140.24219(43)160.22567(55)180.21241(35)200.20072(49)220.19163(23)240.18308(32)320.15833(66)420.13749(40)TABLE I.Results for the magnetization and the susceptibilityIV.ACKNOWLEDGEMENTSThis work was supported by the National Aeronautics and Space Administration under grant no.NAG3-1841.[16]P.Peczak and ndau,Phys.Rev.B47,14260(1963).[17]U.Wolff,Phys.Rev.Lett.62,361(1989).[18]F.R.Brown and T.J.Woch,Phys.Rev.Lett.58,2394(1987).[19]A.M.Ferrenberg and R.H.Swendsen,Phys.Rev.Lett.61,2635(1988);ibid,63,1195(1989).[20]L.S.Goldner,N.Mulders,and G.Ahlers,J.Low Temp.Phys.93,131(1992).11。